Subgroups of the Klein-4 Group

Solution 1:

Hints:

Lagrange's theorem says the only possible sizes of subgroups and orders of elements are $1,2,4$.

The identity element is one of the elements in each of the subgroups, and each element of order $2$ generates a subgroup of order $2$. Are there any elements of order $4$? By thinking about these ideas, you should see how to come up with your list.

The notation $\langle a,b,c,\ldots\mid\ldots \rangle$ is a special one for describing a presentation of a group. It is related to set notation but is not exactly the same.

The letters in the left half denote the supply of symbols that generate the group. Multiplying these symbols just involves using exponents and writing symbols next to each other. For example, $a\cdot b=ab$, $ab\cdot b=ab^2$ and so on. So don't get me wrong: the left half isn't a complete list of what's in the group, it's a list of symbols that generate the group.

The equations in the right half are rules that control the behavor of the multiplication. For example in the Klein $4$ group, one of the rules is that $b^2=1$, so in fact $ab\cdot b=ab^2=a1=a$. These are called relations for this group, since they relate products to each other and control the group operation.

Solution 2:

$G=\{e,a,b,ab\}$ the relation says that every nontrivial element has order $2$.

Thus,$H_1=\{e,a\}$,$H_2=\{e,b\}$,$H_3=\{e,ab\}$,$H_4=\{e\}$,$H_5=G$ are all subgroups.

The notation $<x,y,z..>$ means that group is generated by elements $x,y,z..$ if a group is generated by one elements then it is cylic.In that case, $G=<a,b>$ means every element of $G$ can be written in terms of $a$ and $b$ satisfying the given relation.